Quantum Channel Learning

Abstract

The problem of an optimal mapping between Hilbert spaces $IN$ and $OUT$, based on a series of density matrix mapping measurements $\ddot{I}{}^{(l)}\to {\varrho }^{(l)}$, $l=1\dots M$, is formulated as an optimization problem maximizing the total fidelity $\mathcal{F}={\sum }_{l=1}^M\ddot{I}{‰}^{(l)}F\left({\varrho }^{(l)},{\sum }_s{B}_s\ddot{I}{}^{(l)}{B}_s^{†}\right)$ subject to probability preservation constraints on Kraus operators $B_s$. For $F(\varrho ,\ddot{I}ƒ)$ in the form that total fidelity can be represented as a quadratic form with superoperator $\mathcal{F}={\sum }_s⟨B_s\left|S\right|B_s⟩$ (either exactly or as an approximation) an iterative algorithm is developed. The work introduces two important generalizations of unitary learning: 1. $IN$/$OUT$ states are represented as density matrices. 2. The mapping itself is formulated as a mixed unitary quantum channel $A^{OUT}={\sum }_s|w_s{|}^2{\mathcal{U}}_s{A}^{IN}{\mathcal{U}}_s^{†}$ (no general quantum channel yet). This marks a crucial advancement from the commonly studied unitary mapping of pure states $\ddot{I}{•}_l=\mathcal{U}\ddot{I}{ˆ}_l$ to a quantum channel, what allows us to distinguish probabilistic mixture of states and their superposition. An application of the approach is demonstrated on unitary learning of density matrix mapping ${\varrho }^{(l)}=\mathcal{U}\ddot{I}{}^{(l)}{\mathcal{U}}^{†}$, in this case a quadratic on $\mathcal{U}$ fidelity can be constructed by considering $\sqrt{\ddot{I}{}^{(l)}}\to \sqrt{{\varrho }^{(l)}}$ mapping, and on a quantum channel, where quadratic on $B_s$ fidelity is an approximation -- a quantum channel is then obtained as a hierarchy of unitary mappings, a mixed unitary channel. The approach can be applied to studying quantum inverse problems, variational quantum algorithms, quantum tomography, and more.

The unitary learning from arXiv:2405.10263 is generalized to density matrices and quantum channels. A Schrödinger-like equation is derived for the quantum inverse problem, and a few solutions are presented